Gridding discretization-based multiple stability switching delay search algorithm: The movement of a human being on a controlled swaying bow

PLoS One

Libor Pekař¹  Roman Prokop²  Radek Matušů³ 

[1]Department of Automation and Control Engineering, Faculty of Applied Informatics, Tomas Bata University in Zlín, Zlín, Czech Republic

[2]Department of Mathematics, Faculty of Applied Informatics, Tomas Bata University in Zlín, Zlín, Czech Republic

[3]Regional Research Centre CEBIA-Tech, Faculty of Applied Informatics, Tomas Bata University in Zlín, Zlín, Czech Republic

关键词: Algorithms;    Polynomials;    Interpolation;    System stability;    Control systems;    Transfer functions;    Reflexes;    System instability;   

DOI  :  10.1371/journal.pone.0178950

学科分类：医学（综合）

来源: Public Library of Science

【 摘 要 】

Delay represents a significant phenomenon in the dynamics of many human-related systems—including biological ones. It has i.a. a decisive impact on system stability, and the study of this influence is often mathematically demanding. This paper presents a computationally simple numerical gridding algorithm for the determination of stability margin delay values in multiple-delay linear systems. The characteristic quasi-polynomial—the roots of which decide about stability—is subjected to iterative discretization by means of pre-warped bilinear transformation. Then, a linear and a quadratic interpolation are applied to obtain the associated characteristic polynomial with integer powers. The roots of the associated characteristic polynomial are closely related to the estimation of roots of the original characteristic quasi-polynomial which agrees with the system′s eigenvalues. Since the stability border is crossed by the leading one, the switching root locus is enhanced using the Regula Falsi interpolation method. Our methodology is implemented on—and verified by—a numerical bio-cybernetic example of the stabilization of a human-being′s movement on a controlled swaying bow. The advantage of the proposed novel algorithm lies in the possibility of the rapid computation of polynomial zeros by means of standard programs for technical computing; in the low level of mathematical knowledge required; and, in the sufficiently high precision of the roots loci estimation. The relationship to the direct search QuasiPolynomial (mapping) Rootfinder algorithm and computational complexity are discussed as well. This algorithm is also applicable for systems with non-commensurate delays.

【 授权许可】

CC BY